There are a few different ways you can look at it, but my preference has always been the "do unto others as you would have them do to you" concept. If you were making mistakes, and those mistakes were causing difficulties for someone else, would you prefer they go to your boss or would you rather they just came to you directly?
A concept applied in the military (and it seems to be gradually taking hold in the corporate world is well) is that you always want to handle conflicts at the absolutely lowest level possible. The higher up a problem travels the harder it hits when it finally lands on someone (and the more likely it is to fall back on you).
I would recommend speaking to the intern directly. Don't be a jerk about it, or tell her that she's an incompetent who doesn't deserve to be breathing your air. Guide her.
When you were at the intern level I'm assuming that as an intern there is someone there to help point you in the right direction, pick you up when you do something stupid (come on, we've all done something stupid), and help you become a useful member of the team. Be that leader for her. If you can see her difficulties help her through them. If you can't, help her find someone who can.
Running to tell her boss isn't "snitching" on her, but it's counter productive. It gets involved an individual who doesn't need to be involved (unless the intern is actually being a jerk or doing these things on purpose). If there's a problem, it should be handled right where it's occurring. If the intern doesn't listen, then of course it needs to be escalated. However, keep it diffident. Never become aggressive, hostile or adversarial in any commentary.
I can't imagine you want to be doing the same job your entire life. Ultimately you'll likely want to transition to something supervisory, managerial, etc. Take this opportunity to be the leader and help her out. You might just find that your supervisor/manager takes notice of that kind of initiative and rewards you in some ways for it (beware: the aware for good work is more work). We all need proactive leaders in the workplace.
Note: Edit made due to the concept that while reading originally I did not pick up on the concept that OP is also an intern.